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1.1 root 1: /*
2: * Copyright (c) 1985 Regents of the University of California.
3: * All rights reserved.
4: *
5: * Redistribution and use in source and binary forms are permitted
6: * provided that the above copyright notice and this paragraph are
7: * duplicated in all such forms and that any documentation,
8: * advertising materials, and other materials related to such
9: * distribution and use acknowledge that the software was developed
10: * by the University of California, Berkeley. The name of the
11: * University may not be used to endorse or promote products derived
12: * from this software without specific prior written permission.
13: * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR
14: * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED
15: * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE.
16: *
17: * All recipients should regard themselves as participants in an ongoing
18: * research project and hence should feel obligated to report their
19: * experiences (good or bad) with these elementary function codes, using
20: * the sendbug(8) program, to the authors.
21: */
22:
23: #ifndef lint
24: static char sccsid[] = "@(#)expm1.c 5.3 (Berkeley) 6/30/88";
25: #endif /* not lint */
26:
27: /* EXPM1(X)
28: * RETURN THE EXPONENTIAL OF X MINUS ONE
29: * DOUBLE PRECISION (IEEE 53 BITS, VAX D FORMAT 56 BITS)
30: * CODED IN C BY K.C. NG, 1/19/85;
31: * REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/21/85, 4/16/85.
32: *
33: * Required system supported functions:
34: * scalb(x,n)
35: * copysign(x,y)
36: * finite(x)
37: *
38: * Kernel function:
39: * exp__E(x,c)
40: *
41: * Method:
42: * 1. Argument Reduction: given the input x, find r and integer k such
43: * that
44: * x = k*ln2 + r, |r| <= 0.5*ln2 .
45: * r will be represented as r := z+c for better accuracy.
46: *
47: * 2. Compute EXPM1(r)=exp(r)-1 by
48: *
49: * EXPM1(r=z+c) := z + exp__E(z,c)
50: *
51: * 3. EXPM1(x) = 2^k * ( EXPM1(r) + 1-2^-k ).
52: *
53: * Remarks:
54: * 1. When k=1 and z < -0.25, we use the following formula for
55: * better accuracy:
56: * EXPM1(x) = 2 * ( (z+0.5) + exp__E(z,c) )
57: * 2. To avoid rounding error in 1-2^-k where k is large, we use
58: * EXPM1(x) = 2^k * { [z+(exp__E(z,c)-2^-k )] + 1 }
59: * when k>56.
60: *
61: * Special cases:
62: * EXPM1(INF) is INF, EXPM1(NaN) is NaN;
63: * EXPM1(-INF)= -1;
64: * for finite argument, only EXPM1(0)=0 is exact.
65: *
66: * Accuracy:
67: * EXPM1(x) returns the exact (exp(x)-1) nearly rounded. In a test run with
68: * 1,166,000 random arguments on a VAX, the maximum observed error was
69: * .872 ulps (units of the last place).
70: *
71: * Constants:
72: * The hexadecimal values are the intended ones for the following constants.
73: * The decimal values may be used, provided that the compiler will convert
74: * from decimal to binary accurately enough to produce the hexadecimal values
75: * shown.
76: */
77:
78: #if defined(vax)||defined(tahoe) /* VAX D format */
79: #ifdef vax
80: #define _0x(A,B) 0x/**/A/**/B
81: #else /* vax */
82: #define _0x(A,B) 0x/**/B/**/A
83: #endif /* vax */
84: /* static double */
85: /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */
86: /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */
87: /* lnhuge = 9.4961163736712506989E1 , Hex 2^ 7 * .BDEC1DA73E9010 */
88: /* invln2 = 1.4426950408889634148E0 ; Hex 2^ 1 * .B8AA3B295C17F1 */
89: static long ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)};
90: static long ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)};
91: static long lnhugex[] = { _0x(ec1d,43bd), _0x(9010,a73e)};
92: static long invln2x[] = { _0x(aa3b,40b8), _0x(17f1,295c)};
93: #define ln2hi (*(double*)ln2hix)
94: #define ln2lo (*(double*)ln2lox)
95: #define lnhuge (*(double*)lnhugex)
96: #define invln2 (*(double*)invln2x)
97: #else /* defined(vax)||defined(tahoe) */
98: static double
99: ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */
100: ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */
101: lnhuge = 7.1602103751842355450E2 , /*Hex 2^ 9 * 1.6602B15B7ECF2 */
102: invln2 = 1.4426950408889633870E0 ; /*Hex 2^ 0 * 1.71547652B82FE */
103: #endif /* defined(vax)||defined(tahoe) */
104:
105: double expm1(x)
106: double x;
107: {
108: static double one=1.0, half=1.0/2.0;
109: double scalb(), copysign(), exp__E(), z,hi,lo,c;
110: int k,finite();
111: #if defined(vax)||defined(tahoe)
112: static prec=56;
113: #else /* defined(vax)||defined(tahoe) */
114: static prec=53;
115: #endif /* defined(vax)||defined(tahoe) */
116: #if !defined(vax)&&!defined(tahoe)
117: if(x!=x) return(x); /* x is NaN */
118: #endif /* !defined(vax)&&!defined(tahoe) */
119:
120: if( x <= lnhuge ) {
121: if( x >= -40.0 ) {
122:
123: /* argument reduction : x - k*ln2 */
124: k= invln2 *x+copysign(0.5,x); /* k=NINT(x/ln2) */
125: hi=x-k*ln2hi ;
126: z=hi-(lo=k*ln2lo);
127: c=(hi-z)-lo;
128:
129: if(k==0) return(z+exp__E(z,c));
130: if(k==1)
131: if(z< -0.25)
132: {x=z+half;x +=exp__E(z,c); return(x+x);}
133: else
134: {z+=exp__E(z,c); x=half+z; return(x+x);}
135: /* end of k=1 */
136:
137: else {
138: if(k<=prec)
139: { x=one-scalb(one,-k); z += exp__E(z,c);}
140: else if(k<100)
141: { x = exp__E(z,c)-scalb(one,-k); x+=z; z=one;}
142: else
143: { x = exp__E(z,c)+z; z=one;}
144:
145: return (scalb(x+z,k));
146: }
147: }
148: /* end of x > lnunfl */
149:
150: else
151: /* expm1(-big#) rounded to -1 (inexact) */
152: if(finite(x))
153: { ln2hi+ln2lo; return(-one);}
154:
155: /* expm1(-INF) is -1 */
156: else return(-one);
157: }
158: /* end of x < lnhuge */
159:
160: else
161: /* expm1(INF) is INF, expm1(+big#) overflows to INF */
162: return( finite(x) ? scalb(one,5000) : x);
163: }
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